100 Years of Einstein’s Photoelectric Effect

Pranawa C. Deshmukh and Shyamala Venkataraman Department of Physics

Indian Institute of Technology – Madras Chennai – 600 036

This article is based on a talk PCD was invited to give at a joint function of the Indian Physics Association – Chennai Chapter, The Humboldt Club, Chennai, and the Department of Physics, IIT-Madras on April 20, 2005.Published in the Bulletin of Indian Physics Teachers Association in two parts: September & October Issues of 2006.

Wednesday, November 22, 2006

Section I: 100 years ago, it was Einstein’s “Annus Mirabilis” A century has witnessed the growing impact of three seminal papers written by Albert Einstein in 1905. Indeed it was Einstein’s “Annus Mirabilis” (Miracle Year!); never has so much been written in a single year by anybody that impacted a century of scientific growth! The whole world recognizes this impact, and to commemorate Einstein’s outstanding contributions in 1905, the United Nations declared the year 2005 as the “World Year of Physics” [1]. In India too, the Indian Physics Association and various academic institutions are celebrating the WYP2005 in a big way, including in colleges and schools [2,3]. Albert Einstein was born on March 14, 1879 at Ulm, Germany. Picture 1 shows one of his early photographs [4]. Albert’s father moved their family business of electrical spare parts to Munich, where they did not do too well, whence they moved in 1894 to Milan, Italy. In 1895 Albert Einstein had failed an examination that would have allowed him to study for a diploma as an electrical engineer at Zurich. He worked in a patent office from 1902 to 1909, having been offered only a temporary post when he was first appointed. By 1904 the position was made permanent and he was promoted in 1906, after the “Annus Mirabilis”, to the position of ‘technical expert, second class’ [5]!

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Picture 1: Picture 2: Einstein, at age 3 [4] Young Einstein [6] At his age of 26 in 1905, Einstein had his ‘miracle year’, and least for the fact that during this year he received his Doctorate from the University of Zurich for his dissertation in which he provided a new way of calculating the size of molecules. During this year he made three very significant contributions, each unparalleled in the history of Physics. (i) Einstein’s work on Brownian motion: In 1827, Botanist Robert Brown had observed [7] under the microscope the motion of plant spores floating in water, moving randomly all the time. Albert Einstein made the first satisfactory theoretical treatment of the Brownian motion in 1905. Einstein's theory [8] enabled significant statistical predictions about the motion of particles that are randomly distributed in a fluid. These predictions were later confirmed by experiment and constituted a major breakthrough in this field. (ii) Special Theory of Relativity: The STR is an explanation of the way radiation & matter interact when viewed from different inertial frames of reference. Major upshots of STR [9,10]:

• Physical laws are the same in all inertial reference systems.

• Speed of light in a vacuum is a universal constant for all observers regardless of the motion of the observer or of the source of the light.

• Max velocity attainable is that of light.

• Objects appear to contract in the direction of motion; Rate of moving clock seems to

decrease as its velocity increases.

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• Mass and energy are equivalent and interchangeable.

In fact, the interpretation of the laws of electrodynamics with the relativity principle was at the very basis of the formulation of the STR. [11] (iii) The Photoelectric Effect – the work that was cited in Einstein’s Nobel Prize! Einstein’s paper on the photoelectric effect, like many of his other contributions, turned out to be epoch making. Photoelectric Effect: Our story begins in 1887 with the experiments of Heinrich Hertz. The German Physicist Heinrich Hertz (Picture 3), born 1857, is best known for the fact that he applied Maxwell's theory for the production and reception of radio waves. In recognition of his work, the unit of frequency of a wave - one cycle per second - is named as ‘hertz’. Hertz made some remarkable observations while he was working with the spark-gap generator, which was a primitive radio broadcasting device.

In 1887, Heinrich Hertz and his assistant Philipp Lenard observed that when light was shone on certain substances, the substances gave out cathode rays (electrons); only the number of electrons emitted, and not their energy, increased when the strength (intensity) of the incident light was increased. (Incidentally, a nephew of Heinrich Hertz, Gustav Ludwig Hertz, won the Physics Nobel Prize for 1925 along with James Franck for their discovery of the laws governing the impact of an electron upon an atom.}

Picture 3: Heinrich Hertz [12] The observations of Heinrich Hertz and Philipp Lenard were in contradiction with the then established wave nature of light. Einstein proposed that light can be, rather, considered as particles, and offered a successful explanation of the observed phenomenon. In one of his famous papers published in 1905, Einstein explained the experimental observations of Hertz and Lenard, and established firmly the corpuscular nature of light proposed earlier by Max Planck. Albert Einstein was awarded the Nobel Prize for the year 1921 for his contributions to Theoretical Physics, especially for his explanation of the photoelectric effect. Einstein’s paper on the photoelectric effect, like many of his other contributions, turned out to be epoch making. A very large number of experimental and theoretical developments have stemmed out of it, influencing both pure and applied research. Only just a few of the very many exciting advances in this field that have taken place over a century of intense research will be reviewed in this article, severely prejudiced by our limited experience. Figure 1 shows a schematic arrangement of the experiment carried out by Hertz and Lenard (Picture 4):

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Fig.1: Energy of the ‘photoelectrons’ from the thin metal foil is gauged by varying the

retarding voltage .

Picture 4: Philipp Lenard [13]

White light is sent through a monochromator and is then incident on a thin metal plate. The incident light is absorbed by the metal plate which then ejects electrons from it. These electrons are attracted toward a collector at which they arrive, but only if they have sufficient kinetic energy to get past a grid (a fine wire mesh) to which a retarding potential is applied, as shown in the figure. The kinetic energy of the electrons emitted from the thin metal foil could thus be gauged by varying the retarding voltage. The ‘plate to collector’ electron current, being measured by an ammeter. Lenard made the surprising discovery of a ‘cutoff frequency’.

Lenard’s Observations:

• When a sufficiently negative voltage is set on the grid the electrons do not reach the collector (repelled by the negatively charged grid). Intensity of light had no effect on the energy of ejected electrons (Fig.2).

• There was a threshold frequency below which not a single photoelectron was ejected;

below this frequency, the brightness of light made no difference (Fig.3)!

Lenard was an experimentalist of genius. It is however said that he claimed far more recognition for his investigations than was due. Lenard got many honors, including the 1905 Nobel Prize for his work on cathode rays, but he still felt that he was disregarded. This explains why he frequently attacked some other physicists. Lenard later became a member of Hitler's National Socialist Party which made him the Chief of ‘Aryan’ Physics.

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Fig.2: Fig.3: A sufficiently negative voltage on the The stopping potential is higher grid prevents electrons from making it for larger frequency, and to the collector, regardless of the intensity the two have a linear relationship. of the incident light. Philipp Lenard had systematically recorded all the important observations with regard to the photoelectric effect. It was however Einstein’s paper [14] of 1905 in the distinguished journal ‘Annal der Physic’ in which he interpreted light as ‘particles’ that offered a satisfactory explanation of the photoelectric effect. (An English translation of this paper appeared in ‘American Journal of Physics’ in May, 1965 [15]. The title of this paper is: “Concerning an Heuristic Point of View toward the Emission and Transformation of Light”; it discusses ‘Production of Cathode Rays by Illumination of Solids.’) Until Einstein wrote this paper, it was assumed that electromagnetic radiation traveled as waves. Einstein considered the quantization of light into packets of energy called quanta in the context of propagation of EM energy. Wrote Einstein: “According to the assumption considered here, when a light ray starting from a point is propagated, the energy is not continuously distributed over an ever increasing volume, but it consists of a finite number of energy quanta, localized in space, which move without being divided and which can be absorbed or emitted only as a whole”.

Picture 5: Albert Einstein [16] The Photo-Electric-Effect is named after Einstein, not Lenard. Lenard never forgave Einstein for this!

In essence, Einstein proposed that the incident electromagnetic energy is absorbed as a corpuscle of energy hν by the metal plate. As a result, an electron is ejected from the plate. A part of the energy absorbed goes to overcome the ‘work function’, φ and the residual energy

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is carried by the ejected electron as its kinetic energy. Essentially, Einstein expressed the conservation of energy in the following mathematical form: hν = φ + K.E. (1) As mentioned earlier, the ejected electron is detected at the collector only if its kinetic energy overcomes the retarding potential qV. Thus we get the following relation: hν = φ + qVs (2a) i.e., Vs = (h/q)ν - φ/q. (2b) A graph between the stopping potential Vs and incident frequency of the electromagnetic radiation would thus be a straight line (Fig. 4).

Fig.4: Linear relationship between the stopping potential and the frequency of the EM radiation. Einstein’s interpretation of the photoelectric effect was experimentally verified by Owen W Richardson [17] (who got the Nobel prize for the year 1929 for his study of the work function Φ in thermionic emission). Robert Andrews Millikan (Picture 6) conducted experiments [18] that confirmed that a graph of maximum kinetic energy of the ejected electrons versus frequency is linear. Picture 6: Robert Andrews Millikan [19] Having determined accurately the charge carried by an electron using the elegant and famous "falling drop method"; Millikan carried out the first direct ‘photoelectric’ determination of Planck's constant h (1912-1915) using Einstein’s Eq. 2b. (Besides, Millikan carried out important experiments on the Brownian movements in gases).

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What and where is the photon? By reluctantly introducing a radical new assumption into his mathematics, Planck had explained the Black Body Radiation [20]. He had to assume that energy of the radiation is bundled up in discontinuous parcels, or "quanta". In essence, Planck had discovered the quantum structure of electromagnetic radiation, but he himself did not quite see it that way. Rather, he thought of the new assumption merely as a mathematical trick to obtain the right description of the black body radiation spectral intensity profile [21]. To Planck, his hypothesis was an “…. act of desperation ….”, and he lamented that “….. we will have to live with it …”. The significance of the Planck’s constant h remained for him a mystery, till Picture 7: Einstein explained the photoelectric effect. Planck & Einstein [22] In fact, Thomas Kuhn ([23] Picture 8), well known for his commentary on progress in scientific revelations in terms of successive transitions from one paradigm to another through a process of revolution, has argued [24] as much that it is not to Planck in 1900 but to Einstein in 1905 that we owe the origins of quantum theory. The ‘bundle’ of electromagnetic energy is now known as the ‘photon’, and therefore the phenomenon observed in the Hertz-Lenard experiment, explained by Einstein, as ‘Einstein’s photoelectric effect’. However, the quantum of electromagnetic radiation was not referred to as the ‘photon’ till two decades after Einstein’s seminal paper! In fact, it was the American Chemist, Gilbert Newton Lewis ([25], Picture 9) who coined the word “photon” in a Letter to the Editor to ‘Nature’ [26].

Picture 8: Picture 9: Thomas Samuel Kuhn [23] GILBERT NEWTON LEWIS [25] Scientist, It is often said that no scientist in American history Historian, has contributed more extensively to all fields in Philosopher. Chemistry than Lewis, well known for his explanation of electron-pair bonding in covalent substances and acid-base theory.

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Einstein’s interpretation of the photoelectric effect became a historic building block of the quantum theory. In 1922, Bose ([27], Picture 10) and Einstein published together their famous paper about Bose - Einstein Condensation, in which ‘photon statistics’ were clarified. Nevertheless, ‘Photon’ and ‘Quantum Theory’ both remained enigmatic to him. Einstein remained skeptical about quantum mechanics as God’s scheme for natural phenomena. To record his discomfort with the uncertainty principle of quantum mechanics, Einstein would say: “God does not play dice”, to which Niels Bohr would respond saying “It is not your business to tell God what to do”. The Bohr-Einstein (Picture 11) debates are amongst the most intriguing debates on the foundations of quantum mechanics, and were intensely deliberated upon [28] at the Solvay Conference ([29], Picture 12) of 1927.

Picture 10: Sathyendranath Bose [27] Picture 11: Einstein with Bohr [29] In 1927, Dirac, one of the inventors of Quantum electrodynamics (QED) and who had coined the term boson for particles (like photons) which obey ‘Bose-Einstein’ statistics, treated the quantization of electromagnetic radiation. In 1960, 5 years after Einstein’s death, the laser based on Einstein’s transition mechanism was invented. Nevertheless, Einstein’s discomfort [30,31] about the ‘photon’ and ‘quantum mechanics’ is perhaps borne out by continued worries about ‘what’ the ‘photon’ is, and ‘where’ it is. Richard Feynman is said to have remarked [31] about the ‘whereabouts’ of the photon that “…… nobody knows, and it is best if you try not to think about it …….” Even as there are fresh formulations [32,33] of the photon position operator, it is worth recalling that near the end of his life, Einstein wrote [31]: “All the fifty years of conscious brooding have brought me no closer to the answer to the question: what are light quanta? …..…… Of course, today every rascal thinks he knows the answer, but he is deluding himself.”

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Picture 11: The 1927 Solvay Conference [29] was attended by some of the world's most notable physicists who discussed the newly formulated quantum theory. In this picture are, from left to right: Seated, front row: I. Langmuir, M. Planck, M. Curie, H. A. Lorentz, A. Einstein, P. Langevin, C. E. Guye, C. T. R. Wilson, O. W. Richardson. Middle row: P. Debye, M. Knudsen, W. L. Bragg, H. A. Kramers, P. A. M. Dirac, A. H. Compton, L. V. de Broglie, M. Born, N. Bohr. Standing, last row: A. Piccard, E. Henriot, P. Ehrenfest, E. Herzen, T. De Donder, E. Schroedinger, E. Verschaffelt, W. Pauli, W. Heisenberg, R. H. Fowler, L. Brillouin Section II: Applications of the Photoelectric Effect – Koopmans Theorem and further developments Einstein’s explanation of the photoelectric effect is a statement of conservation of energy wherein the electromagnetic energy is ‘quantized’, all wrapped up in a corpuscular form we now call as ‘photon’. The photon is absorbed if its quantum of energy exceeds the binding energy of an electron in the discrete bound state of the absorber. The excess energy is then carried as kinetic energy of the escaping ‘photoelectron’. Since the atom has discrete quantized states, electrons in bound atomic states have different binding energies, since they cannot all occupy the lowest energy states, being ‘fermions’. The electrons then occupy different available quantized discrete states, one electron per available state, and thus they need photons with different specific energies to be absorbed to eject a photoelectron from respective bound atomic state. The high-energy electromagnetic absorption spectrum then has a saw-tooth form, seen in the Figure 5 [34]. Maximum energy is needed to eject the most tightly bound electron, one in the so called ‘K’ shell. The letter ‘K’ was chosen to label this state, since it is the middle letter of the English alphabets and one did not know at that time if there was any state that may be even more tightly bound than the K shell. Of course, it is now known that the K state is the most tightly bound one.

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In the x-ray absorption spectrum, one thus has the K-edge, the L-edge (seen on resolution as three different edges known as the L1, L2 and L3). The actual position of the absorption edge changes somewhat from one environment of the absorbing atom to another, that is, for an atom in one molecule compared to that in another. Often, an extended absorption fine structure (EXAFS) is seen (Fig.5), and it has detailed information about the physico-chemical information of the absorbing atom.

Fig.5. X-ray absorption coefficient of copper in the region of L and K edges. The box area is

expanded in the inset to show EXAFS and XANES signal [34]. Now, the discrete quantum binding energy of an electron in an atom or ion containing a single electron is given by the eigen-values of the Schrodinger equation. In a many-electron atom/ion, it is more complicated to assign the binding energies that could be used in Einstein’s photoelectric equation. The electron-electron interactions and what are known as many-body ‘correlations’ need to be appropriately dealt with to determine the binding energies. At the lowest level of approximation, the so-called ‘electron correlations’ are ignored. It is assumed that in the many-electron system, the electron wave-functions of other electrons are not affected if one of the electron states is vacant, as would happen if one of the electrons is knocked out in the ‘photoelectric’ process. This approximation is known as the ‘Hartree-Fock frozen orbital approximation’. The binding energy of the photoelectron is then associated with the difference between the energies of the N-electron system, and an (N-1) electron system with a vacancy in the state from which photoelectron is considered to be ejected. The result is mathematically expressed as a very simple equation known as the Koopmans theorem [35]: EHF(N-1,i) – EHF(N) = -εHF,i . (3) The Koopmans’ theorem establishes a direct link between experimentally measurable electron binding energy and the result of the Hartree-Fock mathematical solutions.

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On the experimental front, there have been fascinating developments in high precision spectroscopic techniques based on the photoelectric effect. These developments have been important enough to be recognized by the award of further Nobel prizes to scientists who invented the techniques. An extremely important contribution came from the Swedish

Picture 12: Karl Siegbahn, 1886 - 1978) [36]

physicist Karl Siegbahn of the University of Lund. In 1916 Siegbahn discovered a new group of wavelengths, the M series, in X-ray emission spectra. Most importantly, Siegbahn designed and developed scientific apparatus that is used to determine accurately the wavelengths of X-rays. The experimental technique developed by Karl Siegbahn has emerged as an extremely powerful tool to identify elements and provide detailed information about their physico-chemical environment. Karl Siegbahn (Picture 12, [36a]) was awarded [36b] the Nobel Prize for Physics in 1924 for his important discoveries and for his investigations in X-ray spectroscopy. Later, during the period 1937–64, he served as the director of the Nobel Institute of Physics, Stockholm. Later, Kai Manne Borje [37], son of Karl Siegbahn, exploited the fact that the electron binding energy that appears in Einstein’s photoelectric equation is sensitive to the chemical state of the species, and used this characteristic property as a ‘marker’ for the specific material in which the photon-absorbing atom is embedded. The slightly different binding energy of the electron in an atom in one physico-chemical environment compared to that in another is referred to as ‘chemical shift’. This photoelectron spectroscopic technique is aptly known as ESCA, which stands for Electron Spectroscopy for Chemical Analysis. In an ESCA spectrum, the binding energies of the peaks are characteristic of each element. The position and shape of each peak and binding energy is slightly altered by chemical state of the emitting atom, as illustrated in Figure 6 [38].

Picture 13: Figure 6: C1s photoelectron spectrum Kai Manne Borje Siegbahn [37] peak- fitted to indicate the functional

groups present in a polymethylmethacrylate sample [38].

The strength of XPS is its ability to identify different chemical states. The high precision spectroscopy developed by Kai Manne Borje Siegbahn is thus known as ‘electron spectroscopy for chemical analysis, or ESCA for short. Kai Siegbahn shared the 1981 Nobel Prize in Physics with Nicolaas Bloembergen and Arthur Schawlow. ESCA is also variedly known as PES (Photoelectron Spectroscopy), XPS (X-ray Photoelectron Spectroscopy), UVPES (Ultra-Violet Photoelectron Spectroscopy) etc., depending on details pertaining to which part of the electromagnetic radiation is being used to probe the object. The technique is highly surface specific and therefore has tremendous applications in technology. An overview of the multitude of applications of the photoelectric effect provides an awesome perception of how understanding of atomic and molecular processes has grown [39] over the century since Einstein’s explanation of the phenomenon. Very sophisticated instrumentation, such as cylindrical mirror analyzer, hemispherical analyzers, and angle resolved measurements, etc. [40] have been developed during this period to refine precision measurements. Notably, one must mention pioneering efforts of two research groups: (i) Siegbahn and coworkers [39a,b] at Upasaala, Sweden, and Turner [40] and coworkers at London, U.K. On the theoretical front, advances and applications have both grown significantly. Amongst these are new phenomenology developed about photoelectric effect at high energy; while it has been believed until recently that the independent-particle-approximation provided a satisfactory explanation of the photoelectric process, it is now found [41] that such is not the case. Likewise, it has also been believed till recently that the so-called ‘dipole’ approximation is quite satisfactory up to about 5 KeV above the ionization threshold, it is however now found that non-dipole [42] contributions are important at very low energies as well. Advents in powerful light sources, such as the synchrotron [43], have catalyzed these findings. A very large number of different kinds of experiments primarily based on Einstein’s photoelectric effect are now carried out for research in condensed matter physics. Amongst physical processes that are studied are: vibrationally resolved photoabsorption and

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photoelectron spectra, shape resonance phenomena, double excitations, conjugate satellite transitions, vibronic coupling, determination of angular distribution asymmetry parameters of photoelectrons, partial cross sections, absolute photoabsorption cross sections, electronic de-excitation spectra, etc. The hallmark of quantum mechanical laws of nature is that some measurements are compatible with others, some are not, and to get maximal information about the system, one looks for ‘C.S.C.O.’, i.e. ‘Complete Set of Commuting Operators (Compatible Observables)’. Thus, in addition to measuring the kinetic energy of the photoelectrons given by Einstein’s equation, one measures the angular/spatial distribution of the photoelectrons and their spin-polarization properties [44]. From understanding of fundamental interactions in astrophysical processes [45] to applications in nanotechnology [46], Einstein’s photoelectric effect lends itself as the foundation stone for many a schemes that push the frontiers of science and technology. Acknowledgments: We have attempted only a glimpse at a century of developments based on Einstein’s photoelectric effect. We have perhaps not succeeded in covering even a miniscule of the vast amount of research and development that Einstein’s work has led to. Very many of our teachers, collaborators, colleagues and students have contributed to what has been reported above. We are especially grateful to Professor G. Rangarajan and Professor V.Balakrishnan for their encouragement and support in writing this article. REFERENCES: [1] http://www.physics2005.org/ [2] www.tifr.res.in/~ipa/iyp/summary.htm [3] www.tifr.res.in/~ipa/iyp/index.htm [4] http://www.th.physik.uni-frankfurt.de/~jr/physpiceinstein.html [5a] Brian, Denis. Einstein: A Life. New York: John Wiley, 1996. [5b] Hoffmann, Banesh, with the collaboration of Helen Dukas. Albert Einstein: Creator

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